Question:
Evaluate the following integral:
$\int \frac{5}{\left(x^{2}+1\right)(x+2)} d x$
Solution:
$I=\int \frac{5}{\left(x^{2}+1\right)(x+2)}$
$\frac{5}{\left(x^{2}+1\right)(x+2)}=\frac{A x+B}{x^{2}+1}+\frac{C}{x+2}$
$5=(A x+B)(x+2)+C\left(x^{2}+1\right)$
Equating constants
$5=2 B+C$
Equating coefficients of $x$
$0=2 A+B$
Equating coefficients of $x^{2}$
$0=A+C$
Solving, we get
$A=-1, B=2, C=1$
Thus
$I=\int \frac{-x+2}{x^{2}+1} d x+\int \frac{d x}{x+2}$
$=\int \frac{-\mathrm{xdx}}{\mathrm{x}^{2}+1}+2 \int \frac{\mathrm{dx}}{\mathrm{x}^{2}+1}+\int \frac{\mathrm{dx}}{\mathrm{x}+2}$
$I=-\frac{1}{2} \log \left|x^{2}+1\right|+2 \tan ^{-1} x+\log |x+2|+C$
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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